Proof of Claim REF .
Suppose that \(\mathfrak {C}\)  is minimal prime. Then \(\mathfrak {C}= Age(G)\)  where \(G\)  is a prime graph which is the incomparability graph of a poset coverable by two chains (Theorem REF ). From Theorem 2 of [1]} follows that \(G\)  embeds \(P_\infty \)  or \(H\) . Hence, \(\mathfrak {C}\)  contains \(Age(P_\infty )\)  or \(Age(H)\) . Since \(P_\infty \)  and \(H\)  are prime and \(\mathfrak {C}\)  minimal prime, \(\mathfrak {C}= Age(P_\infty )\)  or \(Age(H)\) .
